The distinction between space and time is evidently quite natural from the point of view of Carrollian transformations. Thus, as we proceed, we shall see that we do not need to assume any exotic `radial quantization' prescription here, contrary to 2D (Euclidean) CFTs. In 2D CFTs, radial ordering on the complex plane is manifestly related to time-ordering [1]} in a theory living on a cylinder \(\mathbb {R}\times {S}^1\) . The 2D CFT plane-to-cylinder map very transparently expresses this fact. To the best of our knowledge, the corresponding \(1+1\) D CCFT `plane-to-cylinder' map existing in the literature [2]} does not have any such physical interpretation without resorting to an analytic-continuation to Euclidean time [3]}. Fortunately, as we shall see, we do not require the service of such a map to establish the operator formalism here. Rather, the temporal \(\theta \) -functions appearing in the OPEs, that can be naturally related to time-ordering, make a very significant contribution in relating the OPEs with the operator commutation relations via a complex contour integral (over \(x\) ) prescription. Due to the richness of complex analytic methods, the operator formalism of \(1+1\) D CCFT will provide us with an extremely powerful computational tool.
